Theorem sotri2 6123

Description: A transitivity relation. (Read 𝐴 ≤ 𝐵 and 𝐵 < 𝐶 implies 𝐴 < 𝐶.) (Contributed by Mario Carneiro, 10-May-2013.)

Hypotheses

Ref	Expression
soi.1	⊢ 𝑅 Or 𝑆
soi.2	⊢ 𝑅 ⊆ (𝑆 × 𝑆)

Assertion

Ref	Expression
sotri2	⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Proof of Theorem sotri2

Step	Hyp	Ref	Expression
1		soi.2	. . . . 5 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
2	1	brel 5724	. . . 4 ⊢ (𝐵𝑅𝐶 → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
3	2	simpld 494	. . 3 ⊢ (𝐵𝑅𝐶 → 𝐵 ∈ 𝑆)
4		soi.1	. . . . . . 7 ⊢ 𝑅 Or 𝑆
5		sotric 5596	. . . . . . 7 ⊢ ((𝑅 Or 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) → (𝐵𝑅𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴𝑅𝐵)))
6	4, 5	mpan 690	. . . . . 6 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵𝑅𝐴 ↔ ¬ (𝐵 = 𝐴 ∨ 𝐴𝑅𝐵)))
7	6	con2bid 354	. . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ((𝐵 = 𝐴 ∨ 𝐴𝑅𝐵) ↔ ¬ 𝐵𝑅𝐴))
8		breq1 5127	. . . . . . 7 ⊢ (𝐵 = 𝐴 → (𝐵𝑅𝐶 ↔ 𝐴𝑅𝐶))
9	8	biimpd 229	. . . . . 6 ⊢ (𝐵 = 𝐴 → (𝐵𝑅𝐶 → 𝐴𝑅𝐶))
10	4, 1	sotri 6121	. . . . . . 7 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
11	10	ex 412	. . . . . 6 ⊢ (𝐴𝑅𝐵 → (𝐵𝑅𝐶 → 𝐴𝑅𝐶))
12	9, 11	jaoi 857	. . . . 5 ⊢ ((𝐵 = 𝐴 ∨ 𝐴𝑅𝐵) → (𝐵𝑅𝐶 → 𝐴𝑅𝐶))
13	7, 12	biimtrrdi 254	. . . 4 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (¬ 𝐵𝑅𝐴 → (𝐵𝑅𝐶 → 𝐴𝑅𝐶)))
14	13	com3r 87	. . 3 ⊢ (𝐵𝑅𝐶 → ((𝐵 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (¬ 𝐵𝑅𝐴 → 𝐴𝑅𝐶)))
15	3, 14	mpand 695	. 2 ⊢ (𝐵𝑅𝐶 → (𝐴 ∈ 𝑆 → (¬ 𝐵𝑅𝐴 → 𝐴𝑅𝐶)))
16	15	3imp231 1112	1 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3931   class class class wbr 5124   Or wor 5565   × cxp 5657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-po 5566  df-so 5567  df-xp 5665

This theorem is referenced by:  supsrlem  11130